(3+1)(1+1) = 8
Q2. How many 3-digit numbers have digit sum equal to 24?
A2. Sets of 3 single digit numbers that add to 24 are {9,9,6} , {9, 8 7} , and {8, 8, 8}. The first can be arranged in 3 ways, the second in 6 ways, and the third in 1 way. So there are 10 ways. Q3. How many strictly increasing sequences of positive integers begin with 1 and end with 7? Two such sequences are 1, 7 and 1,4,6,7.
A3. One possible solution is to systematically list them all. Here is a faster way:
Every sequence starts with 1 and ends with 7. The numbers in between are all the possible subsets of {2, 3, 4, 5, 6}. For any particular subset (e.g. {6, 2, 4}) we don't need to worry about multiple arrangements since we must list the numbers in increasing order (e.g. {2, 4, 6}).
The number of subsets is easy to count using nCr. There are n = 5 elements to choose from and we can choose between r = 0 and r = 5 of them at a time.
So the total is 32.
Q4. In the 3 × 3 grid below, every point is at a unit distance from its nearest horizontal or vertical neighbours. Call a line segment good
if the endpoints of the line segment are grid points, and the line segment contains no grid point other than its endpoints. How many different
good line segments are there?
• • •
• • •
• • •
A4. An easy way to count the number of lines is to figure out how many lines connects with each point, and then divide by two. Each of the 4 corner points is connected with 5 lines. Each of the 4 side points is connected with 7 lines. The middle point is connected with 8 lines. If we sum the lines so far, we have
Of course, now we've counted each line twice (once for each of its endpoints). So the final answer is 28.
Combinatoric Solutions
(Problem Set without solutions:Q1. How many positive intergers divide 2008?
A1. Prime factorization:
(3+1)(1+1) = 8
Q2. How many 3-digit numbers have digit sum equal to 24?
A2. Sets of 3 single digit numbers that add to 24 are {9,9,6} , {9, 8 7} , and {8, 8, 8}. The first can be arranged in 3 ways, the second in 6 ways, and the third in 1 way. So there are 10 ways.
Q3. How many strictly increasing sequences of positive integers begin with 1 and end with 7? Two such sequences are 1, 7 and 1,4,6,7.
A3. One possible solution is to systematically list them all. Here is a faster way:
Every sequence starts with 1 and ends with 7. The numbers in between are all the possible subsets of {2, 3, 4, 5, 6}. For any particular subset (e.g. {6, 2, 4}) we don't need to worry about multiple arrangements since we must list the numbers in increasing order (e.g. {2, 4, 6}).
The number of subsets is easy to count using nCr. There are n = 5 elements to choose from and we can choose between r = 0 and r = 5 of them at a time.
So the total is 32.
Q4. In the 3 × 3 grid below, every point is at a unit distance from its nearest horizontal or vertical neighbours. Call a line segment good
if the endpoints of the line segment are grid points, and the line segment contains no grid point other than its endpoints. How many different
good line segments are there?
• • •
• • •
• • •
A4. An easy way to count the number of lines is to figure out how many lines connects with each point, and then divide by two. Each of the 4 corner points is connected with 5 lines. Each of the 4 side points is connected with 7 lines. The middle point is connected with 8 lines. If we sum the lines so far, we have
Of course, now we've counted each line twice (once for each of its endpoints). So the final answer is 28.